Villagers don't break and replace carrots when their inventory is full of carrots

I thought that when I would make an automatic farm using the villager farming mechanic, that a villager in a brown coat would mine and replace carrots when his inventory was full of carrots.
Instead, he would mine and replace fully grown carrots when he didn't have his inventory filled, but when I filled his inventory with carrots, he stopped.
The auto-farming worked with every other crop in his inventory.

First I would build a farm full of carrots,
Then I would spawn a villager with a brown jacket,
Next I filled his inventory with carrots,
and when I thought he would start farming like the other villagers, he wouldnt